FM radio stations use radio waves with frequencies from 88.0 to 108 MHz to broadcast their signals. Assuming that the inductance has a value of 4.60 x 10^-7 H, determine the range of capacitance values that are needed so the antenna can pick up all the radio waves broadcasted by FM stations.

Question

FM radio stations use radio waves with frequencies from 88.0 to 108 MHz to broadcast their signals. Assuming that the inductance has a value of 4.60 x 10^-7 H, determine the range of capacitance values that are needed so the antenna can pick up all the radio waves broadcasted by FM stations.

_____F (lowest acceptable capacitance)

____F (highest acceptable capacitance)

Answer 1

To determine the range of capacitance values needed for the antenna, we can use the formula for the resonant frequency of an LC circuit.

The resonant frequency is given by:

f = 1 / (2 * π * √(L * C))

In this case, we need to find the range of capacitance values that will allow the antenna to pick up frequencies ranging from 88.0 MHz to 108 MHz.

First, let's convert the frequency range to Hz:

f_min = 88.0 MHz = 88.0 * 10^6 Hz
f_max = 108 MHz = 108 * 10^6 Hz

Now, we can substitute the values into the equation and solve for the capacitance:

f_min = 1 / (2 * π * √(L * C_min))
C_min = 1 / (4 * π^2 * L * f_min^2)

C_min = 1 / (4 * (3.1416)^2 * (4.60 * 10^-7) * (88.0 * 10^6)^2)

Calculating the above expression gives us the value of C_min.

Similarly, for the highest acceptable capacitance:

f_max = 1 / (2 * π * √(L * C_max))
C_max = 1 / (4 * π^2 * L * f_max^2)

C_max = 1 / (4 * (3.1416)^2 * (4.60 * 10^-7) * (108 * 10^6)^2)

Calculating the above expression gives us the value of C_max.

Therefore, by substituting the values of L and f_min/f_max into the respective equations, we can determine the lowest acceptable capacitance (C_min) and the highest acceptable capacitance (C_max) needed for the antenna to pick up all the radio waves broadcasted by FM stations.